S-wave

Time reversibility is an attribute of some stochastic processes and some deterministic processes.

If a stochastic process is time reversible, then it is not possible to determine, given the states at a number of points in time after running the stochastic process, which state came first and which state arrived later.

If a deterministic process is time reversible, then the time-reversed process satisfies the same dynamical equations as the original process (see reversible dynamics); in other words, the equations are invariant or symmetric under a change in the sign of time. Classical mechanics and optics are both time-reversible. Modern physics is not quite time-reversible; instead it exhibits a broader symmetry, CPT symmetry.Template:Fact

Time reversibility generally occurs when, within a process, it can be broken up into sub-processes which undo the effects of each other. For example, in phylogenetics, a time-reversible nucleotide substitution model such as the generalised time reversible model has the total overall rate into a certain nucleotide equal to the total rate out of that same nucleotide.

Time Reversal, specifically in the field of acoustics, is a process by which the linearity of sound waves is used to reverse a received signal; this signal is then re-emitted and a temporal compression occurs, resulting in a reverse of the initial excitation waveform being played at the initial source. Mathias Fink is credited with proving Acoustic Time Reversal by experiment.

Stochastic processes

A formal definition of time-reversibility is stated by Tong in the context of time-series. In general, a Gaussian process is time-reversible. The process defined by a time-series model which represents values as a linear combination of past values and of present and past innovations (see Autoregressive moving average model) is, except for limited special cases, not time-reversible unless the innovations have a normal distribution (in which case the model is a Gaussian process).

A stationary Markov Chain is reversible if the transition matrix {p_ij} and the stationary distribution {π_j} satisfy

$\pi _{i}p_{ij}=\pi _{j}p_{ji},\,$

for all i and j. Such Markov Chains provide examples of stochastic processes which are time-reversible but non-Gaussian.

Time reversal of numerous classes of stochastic processes have been studied including Lévy processes stochastic networks (Kelly's lemma) birth and death processes Markov chains and piecewise deterministic Markov processes.

See also

• Memorylessness
• Kolmogorov’s criterion
• Reversible cellular automaton
• Reversible dynamics
• T-symmetry

References

• Isham, V. (1991) "Modelling stochastic phenomena". In: Stochastic Theory and Modelling, Hinkley, DV., Reid, N., Snell, E.J. (Eds). Chapman and Hall. ISBN 0-412-30390-9 Template:Please check ISBN.
• Tong, H. (1990) Non-linear Time Series: A Dynamical System Approach. Oxford UP. ISBN 0-19-852300-9